For \(E_s\) , standard techniques (see [1]}) show that
\(\frac{\delta E_s(\rho )}{\delta \rho }(\theta ,t)=\mathbb {E}_{x\sim \mu }\left(p^\top _\rho (t;x)f(Z_\rho (t;x),\theta )\right)+e^{-s}|\theta |^2\,,\) 
